Wavelet estimation by Bayesian thresholding and model selection

Many wavelet-based algorithms have been proposed in recent years to solve the problem of function estimation from noisy samples. In particular it has been shown that threshold approaches lead to asymptotically optimal estimation and are extremely effective when dealing with real data. Working under a Bayesian perspective, in this paper we first study optimality of the hard and soft thresholding rules when the function is modelled as a stochastic process with known covariance function. Next, we consider the case where the covariance function is unknown, and propose a novel approach that models the covariance as a certain wavelet combination estimated from data by Bayesian model selection. Simulated data are used to show that the new method outperforms traditional threshold approaches as well as other wavelet-based Bayesian techniques proposed in the literature.